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On a divergent integral

  1. Mar 30, 2012 #1
    greetings . we have the integral :
    [tex] \lim_{T\to \infty }\int_{2-iT}^{2+iT}\frac{(s-1)^{n}}{s}ds[/tex]

    which diverges for every value of n except [itex] n=0 [/itex]
    if we perform the change of variables :

    [tex] s\rightarrow \frac{1}{s}[/tex]

    then :
    [tex]\lim_{T\to \infty }\int_{2-iT}^{2+iT}\frac{(s-1)^{n}}{s}ds=\int_{-i}^{i}\frac{(1-s)^{n}}{s^{n+1}}ds[/tex]

    which converges . am i missing something here , or is this correct !?
  2. jcsd
  3. Mar 30, 2012 #2
    Can you solve it for n=2 using antiderivatives? That is, what is:

  4. Mar 30, 2012 #3


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    Science Advisor

    You need to split the original integration range into two parts at s = 0. Now when you change s to 1/s, you will be able to get the correct integration limits. Note also that the original T becomes 1/T.
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